Function space

In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set X into a vector space have a natural vector space structure given by pointwise addition and scalar multiplication. In other scenarios, the function space might inherit a topological or metric structure, hence the name function space.

Let V be a vector space over a fieldF and let X be any set. The functions X → V can be given the structure of a vector space over F where the operations are defined pointwise, that is, for any f, g : X → V, any x in X, and any c in F, define

When the domain X has additional structure, one might consider instead the subset (or subspace) of all such functions which respect that structure. For example, if X is also vector space over F, the set of linear mapsX → V form a vector space over F with pointwise operations (often denoted Hom(X,V)). One such space is the dual space of V: the set of linear functionalsV → F with addition and scalar multiplication defined pointwise.

If y is an element of the function space C(a,b){\displaystyle {\mathcal {C}}(a,b)} of all continuous functions that are defined on a closed interval [a,b], the norm‖y‖∞{\displaystyle \|y\|_{\infty }} defined on C(a,b){\displaystyle {\mathcal {C}}(a,b)} is the maximum absolute value of y (x) for a ≤ x ≤ b,[2]